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Summary: Fixed-Casimir shooting supplies near-recurrent candidates, but independent integration refinement rejects the saved nonzero checkpoints at the stated closure tolerance.

Independent numerical correction

Follow-up: Accurate cylindrical correction revisits these states with a new independently tested corrector. The rejection below applies to the original saved states, not to every nearby solution.

The earlier claim that this calculation certified a periodic micro-branch is withdrawn. Reintegrating the same saved states and periods without fitting at tighter tolerances gives:

hh original closure norm refined closure norm last endpoint refinement
0.0064 4.25×10−74.25\times10^{-7} 2.41423×10−62.41423\times10^{-6} 1.50×10−91.50\times10^{-9}
0.0096 2.49×10−72.49\times10^{-7} 7.43896×10−67.43896\times10^{-6} 3.24×10−93.24\times10^{-9}
0.0128 5.24×10−75.24\times10^{-7} 5.31450×10−65.31450\times10^{-6} 9.89×10−119.89\times10^{-11}

All three fail the stated 10−610^{-6} closure threshold. Source Casimirs remain preserved to roundoff. The candidates are near-recurrent trajectories, not accepted periodic solutions. Earlier smaller steps have not independently been certified either. This rejects the prior numerical acceptance, not the existence of a nearby periodic orbit.

The refinement uses DOP853 at (rtol,atol)=(10−12,10−14)(\mathrm{rtol},\mathrm{atol})=(10^{-12},10^{-14}) with maximum step T/1200T/1200, then (2.5×10−14,10−15)(2.5\times10^{-14},10^{-15}) with T/2400T/2400. Agreement between those endpoints is an error diagnostic, not a rigorous global error bound or an existence theorem. The force is continuous but its derivative changes at the nuclear-profile boundary; endpoint accuracy must be checked separately from the integrator's local error control.

The original states, parameters, and source-file hashes are preserved in archived candidates. Independent audit checks regenerate the refinement report. The continuation acceptance gate now requires a complete finite integration, fixed Casimirs, and independently refined closure before saving a checkpoint.

Reduced recurrence problem

The exact completed-Coulomb L=3L=3 orbit occupies only coordinates (q0,q1,q3,q4)(q_0,q_1,q_3,q_4) and the corresponding 4×44\times4 skew-source block. This sector remains invariant under both the full-4D Coulomb potential and the finite cylindrical potential supported on (q0,q1)(q_0,q_1). Reducing to it leaves four positions, four momenta, six skew components, and the period.

The shooting residual contains full-state recurrence, the phase condition q1(0)=0q_1(0)=0, and the family condition q0(0)=1q_0(0)=1. Fixed energy is another legitimate family constraint; it was not mathematically inappropriate. Neither constraint guarantees that the desired family exists. The section q1(0)=0q_1(0)=0 is not by itself a radial turning-point condition.

At homotopy h=0h=0, the exact orbit has

T=20.99780573,∥X(T)−X(0)∥=1.86×10−12.T=20.99780573,\qquad \|X(T)-X(0)\|=1.86\times10^{-12}.

Thus the reduced dynamics and recurrence equations reproduce the analytic start.

Why ordinary continuation stalled

For

Vh=(1−h)(−1/∣q∣)+hVrN(∣qvis∣),rN=0.3,V_h=(1-h)(-1/|q|)+hV_{r_N}(|q_{\rm vis}|),\qquad r_N=0.3,

ordinary least-squares steps of h=0.05h=0.05 and 0.0050.005 stagnated with recurrence residuals 0.4590.459 and 0.1510.151 respectively. These are rejected numerical shoots, not evidence that a cylindrical periodic orbit does not exist.

The Jacobian at the exact start is rank deficient, with symmetry and source-invariant structure. Not every unresolved direction has been proved to be a removable gauge direction. Ordinary whole-orbit finite differences misleadingly reported ranks from 8 to 12. This has now been replaced by local complex-step differentiation and simultaneous integration of the 14×1414\times14 fundamental matrix. The local Jacobian agrees with an independent centered derivative to 5.53×10−115.53\times10^{-11}.

The analytic shooting Jacobian has exactly eight resolved singular values,

(94.7368,14.6897,5.94203,5.87715,1.37876,0.928496,0.838421,0.197030), (94.7368,14.6897,5.94203,5.87715,1.37876,0.928496,0.838421,0.197030),

with the other seven at 1.5×10−121.5\times10^{-12} or below. A separately integrated parameter-sensitivity equation gives kernel forcing norm 52.515052.5150. Its component outside the resolved Jacobian range is only 2.70×10−82.70\times10^{-8}, and the minimum-norm tangent has norm 6.209346.20934. Therefore there is no resolved first-order obstruction to continuing the petal orbit toward the cylindrical kernel.

The h=10−4h=10^{-4} tangent predictor has recurrence residual 8.59×10−68.59\times10^{-6}. Truncating newly split soft singular values rather than inverting them gives rank-aware correction residuals 4.87×10−74.87\times10^{-7} and 2.23×10−72.23\times10^{-7}. The corrected period is 20.9909165720.99091657. These are the original, unrefined shooting diagnostics. They do not establish a periodic micro-branch or continuation to the fully cylindrical endpoint h=1h=1.

The unconstrained correction changes the source's quadratic and Pfaffian Casimirs by 2.19×10−32.19\times10^{-3} and 1.08×10−31.08\times10^{-3}. Continuing those values freely is mathematically allowed across a family of source orbits, but is not appropriate if V2 fixes the electron's source coadjoint orbit. Adding both Casimirs raises the resolved shooting rank from 8 to 10. Its analytic first-order equation is still solvable to 2.70×10−82.70\times10^{-8}, with tangent norm 7.323597.32359, so fixed source invariants present no first-order obstruction. The first attempt to impose the Casimirs as extra residuals stalled at 4.49×10−64.49\times10^{-6}. The corrected formulation instead uses

so(4)≅so(3)+⊕so(3)−\mathfrak{so}(4)\cong\mathfrak{so}(3)_+\oplus\mathfrak{so}(3)_-

and parameterizes the fixed coadjoint orbit as the product of the two spheres with the original self-dual and anti-self-dual radii. This removes both Casimir-normal directions identically. The chart reconstructs the initial source to 2.78×10−172.78\times10^{-17} and gives a rank-8-of-13 shooting Jacobian. At h=10−4h=10^{-4}, rank-aware correction reduces the recurrence residual from 1.37×10−51.37\times10^{-5} to 2.44×10−72.44\times10^{-7}, with

T=20.99128584T=20.99128584

and maximum Casimir error 1.78×10−151.78\times10^{-15}. The Casimir parameterization is valid, but this unrefined closure estimate does not certify continuation.

Continuation with the previous residual included in each predictor originally reported the following unrefined values, retained as a diagnostic record:

hh period TT recurrence residual maximum Casimir error
1×10−41\times10^{-4} 20.99128584 2.44×10−72.44\times10^{-7} 1.78×10−151.78\times10^{-15}
2×10−42\times10^{-4} 20.98482005 1.75×10−71.75\times10^{-7} 0
4×10−44\times10^{-4} 20.97204284 2.01×10−72.01\times10^{-7} 4.44×10−164.44\times10^{-16}
8×10−48\times10^{-4} 20.94710312 3.94×10−73.94\times10^{-7} 0
1.6×10−31.6\times10^{-3} 20.89958488 6.03×10−76.03\times10^{-7} 0
3.2×10−33.2\times10^{-3} 20.81331458 6.31×10−76.31\times10^{-7} 8.88×10−168.88\times10^{-16}
6.4×10−36.4\times10^{-3} 20.67292125 4.25×10−74.25\times10^{-7} 8.88×10−168.88\times10^{-16}
9.6×10−39.6\times10^{-3} 20.56750468 2.49×10−72.49\times10^{-7} 4.44×10−164.44\times10^{-16}
1.28×10−21.28\times10^{-2} 20.48965107 5.24×10−75.24\times10^{-7} 4.44×10−164.44\times10^{-16}

The optimizer reached h=0.0128h=0.0128 using an intermediate h=0.0096h=0.0096 candidate, but the independent check above rejects its accuracy. Neither an accepted 1.28% continuation branch nor a branch termination follows. First-order solvability at a rank-deficient start also does not, alone, imply nonlinear continuation.

Required continuation method

First correct candidates using integration accuracy independently adequate for the desired closure tolerance. Pseudo-arclength continuation is a candidate improvement, not a guaranteed remedy. Multiple-shooting nodes can be placed at the six cylindrical-support encounters, where the profile changes curvature most rapidly. The existing fixed SO(4)SO(4) source chart already uses self-dual and anti-self-dual two-spheres and removes the two Casimir-normal directions. Time phase, amplitude/arclength, and the remaining visible/unobserved rotation gauges must also be fixed explicitly. Only after reaching h=1h=1 should the full reciprocal moving-nucleus system be continued from the fixed-centre solution and its Floquet multipliers computed.

Reproduction

cylindrical_petal_continuation_checks.py reproduces the exact start, verifies its local analytic Jacobian, integrates the monodromy and parameter sensitivity, tests first-order solvability, and performs two rank-aware corrections of the nonzero microstep.

fixed_casimir_petal_continuation_checks.py constructs the two-sphere source chart and attempts continuation. It must reject a candidate that fails its independent integration-quality gate.